Note that the empirical risk approximation \(\hat{R}\)  in (REF ) can be interpreted as an approximation of the data distribution \(\mathfrak {D}\)  by a discrete distribution that places atoms on each data point. In that sense, \(\hat{R}_\text{aug}\)  in (REF ) can be thought of as the Vicinal Risk Minimization [1]} counterpart of (REF ), in which the atoms on \(\mathbf {x}_i\)  are replaced by a local distribution over the transformed samples \(g\mathbf {x}_i\) , i.e.,
\(\hat{R}_{\text{aug}}(f_{\theta }) = \frac{1}{N}\sum _{i=1}^N \int \ell \left(h(g\mathbf {x}_i), y_i\right) d P(g\mathbf {x}_i),\) 
